Title AN INVITATION TO THE SIMILARITY PROBLEMS :AFTER PISIER(Operator Space Theory and its Applications)

Author(s) OZAWA, NORUTAKA

Citation 数理解析研究所講究録 (2006), 1486: 27-40

Issue Date 2006-04

URL http://hdl.handle.net/2433/58153

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

AN INVITATION TO THE SIMILARITY PROBLEMS(AFTER PISIER)

NARUTAKA OZAWA (小沢登高, 東大数理)

ABSTRACT. This note is intended as a handout for the minicourse given in RIMSworkshop $‘(\mathrm{O}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}$ Space Theory and its Applications” on January 31, 2006.

1. THE SIMILARITY PROBLEMS

1.1. The similarity problem for continuous homomorphisms. In this note,we mainly consider unital $\mathrm{C}^{*}$-algebras and unital (not necessarily $*$-preserving)homomorphisms for the sake of simplicity. Let $A$ be a unital C’-algebra and$\pi:Aarrow \mathrm{B}(\mathcal{H})$ be a unital homomorphism with $||\pi||<\infty$ . We say that $\pi$ issimilar to $\mathrm{a}*$-homomorphism if there exists $S\in \mathrm{G}\mathrm{L}(\mathcal{H})$ such that $\mathrm{A}\mathrm{d}(S)\circ\pi$ isa $*$-homomorphism. Here, $\mathrm{G}\mathrm{L}(\mathcal{H})$ is the set of invertible element in $\mathrm{B}(\mathcal{H})$ and$\mathrm{A}\mathrm{d}(S)(x)=SxS^{-1}$ .Similarity Problem A (Kadison 1955). Is every continuous homomorphismsimilar to $\mathrm{a}*$-homomorphism?

We note that a homomorphism $\pi$ is a $*$-homomorphism iff $||\pi||=1$ , since anelement $x\in \mathrm{B}(\mathcal{H})$ is unitary iff $||x||=||x^{-1}||=1$ . We say $A$ has the similarityproperty (abbreviated as $(\mathrm{S}\mathrm{P})$ ) if every unital continuous homomorphism from $A$

into $\mathrm{B}(\mathcal{H})$ is similar to a $*$-homomorphism. Do we really need the assumptionthat $\pi$ is continuous? That is another problem. Indeed, the subject of automaticcontinuity is extensively studied in Banach algebra theory, and it is known that theexistence of a discontinuous homomorphism from a $\mathrm{C}^{*}$-algebra into some Banachalgebra is independent of (ZFC). As far as the author knows, it is not knownwhether or not the automatic continuity of a homomorphism between $\mathrm{C}$“-algebras(say, with a dense image) is provable within (ZFC).

Similarity Problem A is equivalent to several long-standing problems in $\mathrm{C}^{*}$ , vonNeumann and operator theories. Among them is the Derivation Problem;Derivation Problem. Is every derivation 6: $Aarrow \mathrm{B}(\mathcal{H})$ inner?

Let $A\subset \mathrm{B}(\mathcal{H})$ be a (unital) C’-algebra. A derivation 6: $Aarrow \mathrm{B}(\mathcal{H})$ is a linearmap which satisfies the derivative identity $\delta(ab)=\delta(a)b+a\delta(b)$ . The celebratedtheorem of Kadison and Sakai is that every derivation into $A”$ is inner. We recall

Date: February 20, 2006.

数理解析研究所講究録1486巻 2006年 27-40 27

NARUTAKA OZAWA

that $\delta:Aarrow \mathrm{B}(\mathcal{H})$ is said to be inner if there exists $T\in \mathrm{B}(\mathcal{H})$ such that$\forall a\in A$ $\delta(a)=\delta_{T}(a):=Ta-aT$ .

It is known that every derivation is automatically continuous (Ringrose). We say$A$ has the $(\mathrm{D}\mathrm{P})$ if any derivation $\delta:Aarrow \mathrm{B}(\mathcal{H})$ , for any faithful $*$-representation$A\subset \mathrm{B}(\mathcal{H})$ , is inner.

Theorem 1.1 (Kirchberg 1996). Let $A$ be a unital $\sigma$ -algebra. Then $A$ has the$(\mathrm{S}\mathrm{P})$ iff $A$ has the $(\mathrm{D}\mathrm{P})$ .

The easier implication $(\mathrm{S}\mathrm{P})\Rightarrow(\mathrm{D}\mathrm{P})$ (which precedes Kirchberg) follows fromthe following lemma.

Lemma 1.2. Let $A\subset \mathrm{B}(\mathcal{H})$ be a unital $\sigma$ -algebra and $\delta:Aarrow \mathrm{B}(\mathcal{H})$ be a deriva-tion. Then the homomorphism $\pi:Aarrow \mathrm{M}_{2}(\mathrm{B}(\mathcal{H}))$ defined by

$\pi(a)=(0a\delta(a)a)$

is similar to $a*$ -homomorphism iff $\delta$ is inner.

Proof. We first observe that $\pi$ is indeed a homomorphism since 6 is a derivation.If $\delta=\delta_{T}$ , then we have

$\pi(a)=$and $\pi$ is similar to a $*$-homomorphism $\mathrm{i}\mathrm{d}_{A}\oplus \mathrm{i}\mathrm{d}_{A}$ . We now suppose that $\sigma(a)=$

$S\pi(a)S^{-1}$ is $\mathrm{a}*$-homomorphism. Let $D=S’ S$ . Since

Il $S^{-1}||^{2}\langle D\xi,\xi\rangle=||S^{-1}||^{2}||S\xi||^{2}\geq||\xi||^{2}$,

we have $D\geq||S^{-1}||^{-2}$ . Since $\sigma \mathrm{i}\mathrm{s}*$-preserving, we have

$D\pi(a)=S^{*}\sigma(a)S=(S^{*}\mathrm{a}(a^{*})S)^{*}=\pi(a^{*})^{*}D$

for every $a\in A$ . Developing the equation, we get

$(a0\delta(a)a)=$$\mathrm{L}\mathrm{o}\mathrm{o}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{h}\mathrm{e}(1, 1)- \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{y},\mathrm{w}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}D_{11}a=aD_{11}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}a\in A.\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}D_{11}\geq||S^{-1}||^{-2},\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}1\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}D_{11}^{-1}\in A’\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}||D_{11}^{-1}||\leq||S^{-1}||^{2}.\mathrm{L}\mathrm{o}\mathrm{o}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{h}\mathrm{e}$

$(2,1)$-entry, we have$D_{11}\delta(a)+D_{12}a=aD_{12}$ .

It follows that $\delta=\delta_{T}$ for $T=-D_{11}^{-1}D_{12}$ with $||T||\leq||S||^{2}||S^{-1}||^{2}$ . $\square$

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SIMILARITY PROBLEMS AFTER PISIER

1.2. Known cases and open cases. The important result of Haagerup (1983)is that a continuous homomorphism $\pi:Aarrow \mathrm{B}(\mathcal{H})$ admitting a finite cyclic subset(i.e., there exists a finite subset $F\subset \mathcal{H}$ such that $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{\pi(a)\xi : a\in A, \xi\in F\}$ isdense in $\mathcal{H}$ ), is inner. This does not finish the similarity problem since we cannotdecompose a general ( $\mathrm{n}\mathrm{o}\mathrm{n}*$-preserving) representation into a direct sum of cyclicrepresentations.

Theorem 1.3. The following $\sigma$ -algebras have the $(\mathrm{S}\mathrm{P})$ .(1) Nuclear O-algebras.(2) $\sigma$ -algebras $withov,t$ tracial states (Haagerup).(3) Type $\mathrm{I}\mathrm{I}_{1}$ factors with the property $(\Gamma)$ (Chnstensen).

We note that one may reduce Similarity problem A (or derivation problem) forC’-algebras to that for type $\mathrm{I}\mathrm{I}_{1}$ factors by considering the second dual, then con-sidering the type decomposition and direct integration. We do not know whetheror not the von Neumann algebras $\mathcal{L}\mathrm{F}_{2}$ and $\prod_{n=1}^{\infty}\mathrm{M}_{n}$ have the $(\mathrm{S}\mathrm{P})$ . We suspectthat $\prod_{n=1}^{\infty}\mathrm{M}_{n}$ should be a counterexample.

1.3. The similarity problem for group representations. We only considerdiscrete groups. Let $\Gamma$ be a discrete group and $C^{*}\Gamma$ be the full group C’-algebra.We regard $\Gamma$ as the corresponding subgroup of unitary elements in $C^{*}\Gamma$ . Everycontinuous homomorphism $\pi:C^{*}\Gammaarrow \mathrm{B}(\mathcal{H})$ gives rise to a uniforiy bounded(abbreviated as $\mathrm{u}.\mathrm{b}.$ ) representation of $\Gamma$ on $\mathcal{H};\pi:\Gammaarrow \mathrm{G}\mathrm{L}(\mathcal{H})$ is a group homo-morphism such that $|| \pi||:=\sup_{s\in\Gamma}||\pi(s)||<\infty^{1}$ . Obviously, the homomorphism$\pi:C^{*}\Gammaarrow \mathrm{B}(\mathcal{H})$ is similar to $\mathrm{a}*$ -homomorphism iff the representation $\pi_{|\Gamma}$ is uni-tarizable (i.e., $\exists S\in \mathrm{G}\mathrm{L}(\mathcal{H})$ such that Ad(S) $\circ\pi_{|\Gamma}$ is a unitary representation).

Theorem 1.4 (Diximier 1950). Let $\Gamma$ be an amenable group. Then, every $u.b$ .representation of $\Gamma$ is unitarizable. More precisely, if $\pi:\Gammaarrow \mathrm{G}\mathrm{L}(\mathcal{H})$ is a $u.b$ .representation, then there $e$ rists $S\in \mathrm{G}\mathrm{L}(\mathcal{H})\cap \mathrm{v}\mathrm{N}(\pi(\Gamma))$ with $||S||||S^{-1}||\leq||\pi||^{2}$

such that Ad(S) $\circ\pi$ is unitary.

Proof. Let $\Gamma$ be amenable and $\pi:\Gammaarrow \mathrm{G}\mathrm{L}(\mathcal{H})$ be a $\mathrm{u}.\mathrm{b}$ . representation. Let $F_{n}\subset\Gamma$

be a $\mathrm{F}\emptyset \mathrm{l}\mathrm{n}\mathrm{e}\mathrm{r}$ net. Since $\pi$ is $\mathrm{u}.\mathrm{b}.$ , the set $|F_{n}|^{-1} \sum_{s\in F_{n}}\pi(s)^{*}\pi(s)\in \mathrm{v}\mathrm{N}(\pi(\Gamma))$ has aweak’-accumulation point. Since the accumulation point is positive, we kt $S$ bethe the square root of it. Then, we have

$||S \xi||^{2}=\lim_{n}\frac{1}{|F_{n}|}\sum_{s\in F_{n}}||\pi(s)\xi||^{2}$ ,

and hence $||\pi||^{-1}\leq S\leq||\pi||$ and $||S\pi(s)\xi||=||S\xi$ II for every $s\in\Gamma$ and $\xi\in \mathcal{H}$ . Itfollows that $||\mathrm{A}\mathrm{d}(S)\circ\pi||=1$ and hence Ad(S) $\circ\pi$ is unitary. $\square$

1This notation may cause confusion since the value $||\pi||$ is not same as $||\pi:C^{*}\Gammaarrow \mathrm{B}(\mathcal{H})||$ .

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If one employ the fact that a nuclear C’-algebra is amenable as a Banach algebra(Haagerup 1983), then we can adopt the above proof to the case of nuclear $\mathrm{C}^{*}-$

algebras. We say $\Gamma$ is unitarizable if every $\mathrm{u}.\mathrm{b}$ . representation of $\Gamma$ is unitarizable.Pisier $(2004, 2005)$ proved that if $\Gamma$ is unitarizable and in addition that the similar-ity $S$ can be chosen so that (i) $S\in \mathrm{G}\mathrm{L}(\mathcal{H})\cap \mathrm{v}\mathrm{N}(\pi(\Gamma))$ , or (ii) $||S||||S^{-1}$ Il $\leq||\pi||^{2}$ ,then $\Gamma$ is amenable. However, the following is still open.Similarity Problem B. Is every unitarizable group amenable?

Theorem 1.5. The free group $\mathrm{F}_{\infty}$ on countably many generators is not unitariz-able.

Proof. We denote by $|t|$ the word length of $t\in$ $\mathrm{F}_{\infty}$ , by $\mathbb{C}\mathrm{F}_{\infty}$ the space of all finitelysupported $\mathbb{C}- \mathrm{v}\mathrm{a}\mathrm{l}\iota \mathrm{l}\mathrm{e}\Lambda$ fimctions on $\mathrm{F}_{\infty}$ , and by $\lambda(,9)$ the left translation operator by$s$ on $\ell_{\infty}\Gamma$ (and its subspaces). Let $B:\mathbb{C}\mathrm{F}_{\infty}arrow\ell_{\infty}\mathrm{F}_{\infty}$ be the linear map defined by

$B \delta_{\mathrm{t}}=\sum\{\delta_{t’} : |t^{-1}t’|=1, |t’|=|t|+1\}$ ,

i.e., $B\delta_{t}$ is the characteristic function of those points which are just one-step aheadof $t$ (looking from $e$). Then, for every $s\in \mathrm{F}_{\infty}$ , we have

$(B\lambda(s)-\lambda(s)B)\delta_{t}=\{$$0$ if $|s|\neq|st|+|t^{-1}|$

$\delta_{s(|st|+1)}-\delta_{s(|st|-1)}$ if $|s|=|st|+|t^{-1}|$

where $s(k)$ is the unique element such that $|s(k)|=k$ and $|s|=|s(k)|+|s(k)^{-1}s|$ .Hopefully, the figures below explain the above equation. It follows that we may

FIGURE 1. $|s|\neq|st|+|t^{-1}|$ FIGURE 2. $|s|=|st|+|t^{-1}|$

view $D(s)=B\lambda(s)-\lambda(s)B$ as an element in $\mathrm{B}(\ell_{2}\mathrm{F}_{\infty})$ with $||D(s)||=2$ . Thus,$D:\mathrm{F}_{\infty}arrow \mathrm{B}(\ell_{2}\mathrm{F}_{\infty})$ is a $\mathrm{u}.\mathrm{b}$ . derivation; $D(st)=D(s)\lambda(t)+\lambda(s)D(t)$ . It is nothard to show that $D$ is not inner, i.e., there is no $B_{0}\in \mathrm{B}(\ell_{2}\mathrm{F}_{\infty})$ such that $B-B_{0}$

commutes with every $\lambda(s)$ (in $L(\mathbb{C}\mathrm{F}_{\infty},$ $P_{\infty}\mathrm{F}_{\infty})$). We define a $\mathrm{u}.\mathrm{b}$ . representation$\pi:\mathrm{F}_{\infty}arrow \mathrm{M}_{2}(\mathrm{B}(\ell_{2}\mathrm{F}_{\infty}))$ by

$\pi(s)=(\lambda(s)0$ $D(s)\lambda(s))$ .

We conclude the proof by using the fact, which is proved in the same way toLemma 1.2, that $\pi$ is similar $\mathrm{t}\mathrm{o}*$-homomorphism only if $D$ is inner. $\square$

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We observe that a subgroup of a unitarizable group is again unitarizable thanksto the fact that the induction of a $\mathrm{u}.\mathrm{b}$ . representation is again $\mathrm{u}.\mathrm{b}$ . (and a littlemore effort). Hence a counterexample (if any) to Similarity Problem $\mathrm{B}$ has to be anon-amenable group which does not contain $\mathrm{F}_{2}$ as a subgroup. Do you think thismight be a good time to stop chasing the problem?

2. ISOMORPHIC CHARACTERIZATION OF INJECTIVITY2.1. A free Khinchine inequality. Let $\Gamma$ be a discrete group and LF be itsgroup von Neumann algebra. By definition, the map

$\mathcal{L}\Gamma\ni\lambda(f)rightarrow f=\lambda(f)\delta_{e}\in\ell_{2}\Gamma$

is contractive. For which operator space structure on $\ell_{2}\Gamma$ , does the above mapcompletely bounded? We briefly review the column and row Hilbert space struc-tures. Let $\mathcal{H}$ be a Hilbert space. When it is viewed as a column vector space, wesay it is a column Hilbert space and denote it by $\mathcal{H}_{C}$ , i.e., $\mathcal{H}_{C}=\mathrm{B}(\mathbb{C}, \mathcal{H})$ as anoperator space. For any finite sequence2 $(x_{i})_{i}$ in $\mathrm{B}(H)$ and orthonormal vectors$\xi_{1},$

$\ldots,$$\xi_{n}\in \mathcal{H}$ , we have

$||(x_{i})_{i}||_{c:}=|| \sum_{i}x:\otimes\xi_{i}||_{\mathrm{B}(H)\otimes \mathcal{H}_{O}}=||||=||\sum_{i}x:.x:||^{1/2}$ .

Likewise, we define the row Hilbert space as $\mathcal{H}_{R}=\mathrm{B}(\overline{\mathcal{H}}, \mathbb{C})$ , where $\overline{\mathcal{H}}$ is theconjugate Hilbert space of $\mathcal{H}$ . For any finite sequence $(x_{i})$ in $\mathrm{B}(H)$ and orthonormalvectors $\xi_{1},$

$\ldots,$$\xi_{n}\in \mathcal{H}$ , we have

$||(x_{i})_{i}||_{R}:=|| \sum_{i}x_{i}\otimes\xi_{i}||_{b(H)\otimes \mathcal{H}_{R}}=||(x_{1}$$x_{2}$ ...

$)||=|| \sum_{i}x_{i}x_{i}^{*}||^{1/2}$ .

We regard the following lemma trivial and use it without referring it.

Lemma 2.1. For any finite sequences $(a_{i})_{i}$ and $(b_{i})_{i}$ in $\mathrm{B}(\mathcal{H})$ , we have

$|| \sum_{i}a_{\}b_{i}||\leq||(a_{i})_{i}||_{R}||(b_{i})_{i}||_{C}$ .

In particular, $|| \sum a_{i}\otimes b_{i}||\leq\min\{||(a_{i})_{i}||_{R}||(b_{i})_{i}||_{C}, ||(a_{i})_{1}||_{C}||(b_{1})_{1}||_{R}\}$ .We define $\mathcal{H}_{C\cap R}=\{\xi\oplus\xi\in \mathcal{H}_{C}\oplus \mathcal{H}_{R} : \xi\in \mathcal{H}\}$ .

Proposition 2.2. The map$L\Gamma\ni\lambda(f)\mapsto f\in(p_{2}\Gamma)_{C\cap R}$

is completely contractive.

2A finite sequence is a sequence of vectors such that all but finitely many are zero

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Proof. We view $\delta_{e}\in \mathrm{B}(\mathbb{C}, \ell_{2}\Gamma)$ and $\delta_{e}^{*}\in \mathrm{B}(\overline{l_{2}\Gamma}, \mathbb{C})$. Since $f=\lambda(f)\delta_{e}\in \mathrm{B}(\mathbb{C},l_{2}\Gamma)$ ,the above map is a complete contraction into $\mathcal{H}_{C}$ . Since $f=\delta_{e}^{*}\overline{\lambda}(f)\in \mathrm{B}(\overline{l_{2}\Gamma}, \mathbb{C})$ ,the above map is a complete contraction into $\mathcal{H}_{R}$ as well. $\square$

We simply write $C\cap R$ for $(\ell_{2})_{C\cap R}$ and $\{\theta_{i}\}$ for a fixed orthonormal basis for$C\cap R$ . For instance, we can take $\theta_{i}=e_{i1}\oplus e_{1i}\in \mathrm{B}(\ell_{2})\oplus \mathrm{B}(\ell_{2})$ . For a finitesequence $(x_{\mathfrak{i}})_{1}$ in $\mathrm{B}(\mathcal{H})$ , we set

$||(x_{i})_{i}||_{C\cap R}=|| \sum_{i}x_{\mathfrak{i}}\otimes\theta_{i}||_{u(\mathcal{H})\otimes(C\cap R)}\sim=\max\{||(x_{i})_{i}.||_{C}, ||(x_{i})_{i}||_{R}\}$ .

The following is the rudiment of hee Khinchine inequalities.

Theorem 2.3 (Haagerup and Pisier 1993). Let $\mathrm{F}_{\infty}$ be the free group on countablegenerators, $S=\{s_{i}\}\subset \mathrm{F}_{\infty}$ be the standard set of free generators and

$E_{\lambda}=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{I}1}\{s_{i}\}\subset \mathcal{L}\mathrm{F}_{\infty}$

be an operator subspace. Then, the map$\Phi:C\cap R\ni\theta_{\mathfrak{i}}\mapsto\lambda(s_{i})\in L\mathrm{F}_{\infty}$

is completely bounded with $||\Phi||_{\mathrm{c}\mathrm{b}}\leq 2$ . In particular, the projection $Q$ from $\mathcal{L}\mathrm{F}_{\infty}$

onto $E_{\lambda}$ , defined by

$Q:\mathcal{L}\mathrm{F}_{\infty}\ni\lambda(s)\mapsto\{$

$\lambda(s)$ if $s\in S$

$0$ if $s\not\in S$’

is completely bounded with $||Q||_{\mathrm{c}\mathrm{b}}\leq 2$ .

Proof. For each $i$ , let $\Omega_{i}^{\pm}\subset \mathrm{F}_{\infty}$ be the subsets of all reduced words which beginswith respectively $s_{i}^{\pm 1}$ , and $P_{\mathfrak{i}}^{\pm}\in \mathrm{B}(\ell_{2}\mathrm{F}_{\infty})$ be the orthogonal projection onto $l_{2}\Omega_{i}^{\pm}$ .Then, for each $i$ , we have

$\lambda(s_{i})=\lambda(s_{i})P_{i}^{-}+\lambda(s_{i})(1-P_{1}^{-})=\lambda(s_{i})P_{i}^{-}+P_{i}^{+}\lambda(s_{t})$.Therefore for any finite sequence $(x_{i})_{i}\subset \mathrm{B}(H)$ , we have

$|| \sum_{i}x_{i}\otimes\lambda(s_{i})P_{t}^{-}||_{1\mathrm{B}(H\otimes\ell_{2}\mathrm{p}_{\infty})}\leq||(x_{\mathfrak{i}})_{i}||_{R}||(\lambda(s_{\mathfrak{i}})P_{i}^{-})_{i}||_{C}\leq||(x_{i})_{i}||_{R}$

since $||( \lambda(s_{i})P_{i}^{-})_{i}||_{C}=||\sum_{:}P_{i}^{-}||^{1/2}=1$ . Likewise, we have

$|| \sum_{i}x_{i}\otimes P_{\mathfrak{i}}^{+}\lambda(s_{\mathfrak{i}})||_{\mathrm{B}(H\otimes\ell_{2}\mathrm{F}_{\infty})}\leq||(x_{i})_{1}||_{C}||(P_{i}^{+}\lambda(s_{i})):||_{R}\leq||(x_{\mathfrak{i}})_{i}||c$.

It follows that

II $\sum_{:}x_{i}\otimes\lambda(s_{i})||_{\mathrm{I}\mathrm{B}(H\emptyset\ell_{2}\mathrm{F}_{\infty})}\leq 2||(x_{i})_{\mathfrak{i}}||_{C\cap R}=2||\sum_{i}x_{i}\otimes\theta_{\mathfrak{i}}||$.

This means that $||\Phi||_{\mathrm{c}\mathrm{b}}\leq 2$ . The second assertion follows from Proposition 2.2. $\square$

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Remark 2.4. The above property of $\mathcal{L}\mathrm{F}_{\infty}$ is related to the fact that $\mathcal{L}\mathrm{F}_{\infty}$ is notinjective. We simply write $E_{n}$ for $(\ell_{2}^{n})_{c\mathrm{n}R}$ . Thus

$E_{n}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{e_{i1}\oplus e_{1i} : i=1, \ldots, n\}\subset \mathrm{M}_{n}\oplus \mathrm{N}\mathrm{I}_{n}$ .It is known that $E_{n}$ is far from injective, i.e., any projection from $\mathrm{M}_{n}\oplus \mathrm{M}_{n}$ onto $E_{n}$

has cb-norm $\geq\frac{1}{2}(\sqrt{n}+1)$ . It follows that if $M$ is an injective von Neumann algebra,then any maps $\alpha$ : $E_{n}arrow M$ and 5: $Marrow E_{n}$ with $\beta 0\alpha=\mathrm{i}\mathrm{d}_{E_{\hslash}}$ satisfy $||\alpha||_{\mathrm{c}\mathrm{b}}||\beta||_{cb}\geq$

I $(\sqrt{n}+1)$ . It is conjectured by Pisier(?) that for any non-injective von Neumannalgebra $M$ , there exist sequences of maps $\alpha_{n}$ : $E_{n}arrow M$ and $\beta_{n}$ : $Marrow E_{n}$ suchthat $\beta_{n}\circ\alpha_{n}=\mathrm{i}\mathrm{d}_{E_{n}}$ and $\sup||\alpha_{n}||_{\mathrm{c}\mathrm{b}}||\beta_{n}||_{cb}<\infty$ . An affirmative answer would solveseveral problems around operator spaces (e.g., whether existence of a boundedlinear projection from $\mathrm{B}(\mathcal{H})$ onto $M$ implies injectivity of $M.$ ) A negative answerwould lead to a non-injective type $\mathrm{I}\mathrm{I}_{1}$ factor which does not contain $\mathcal{L}\mathrm{F}_{2}$ .2.2. Isomorphic characterization of injective von Neumann algebras. Fora finite sequence $(x_{i})_{i}$ in $\mathrm{B}(\mathcal{H})$ , we set

$||(x_{i})_{i}||_{C+R}=||\Phi:C\cap R\ni\theta_{i}\mapsto x_{i}\in \mathrm{B}(\mathcal{H})||_{\mathrm{c}\mathrm{b}}$.We say that a von Neumann algebra $M$ has the property $(\mathrm{P})^{3}$ if there exists aconstant $C_{M}>0$ with the following property; For any finite sequence $(x_{i})_{i}$ in $M$

with $||(x_{i})_{i}||c+R\leq 1$ , there exist finite sequences $(a_{i})_{i}$ and $(b_{i})_{i}$ in $M$ such that$||(a_{i})_{i}||_{C}\leq C_{M},$ $||(b_{i})_{1}||_{R}\leq C_{M}$ and $x_{i}=a_{1}+b_{:}$ for every $i$ .

Theorem 2.5 (Pisier 1994). A von Neumann algebra $M$ is injective iff it has theproperty (P).

The “if” part requires several lemmas, and we first prove the “only if” part.Let $M$ be an injective von Neumann algebra and consider a complete contraction$\Phi:C\cap R\ni\theta_{i}$ ト\rightarrow xi $\in M$ . Since $M$ is injective, this map extends to a completecontraction $\tilde{\Phi}$ : $C\oplus Rarrow M$ , where $C=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\{e_{\mathrm{t}1}\}$ and $R=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\{e_{1i}\}$ . Then$a:=\tilde{\Phi}(0\oplus e_{1i})$ and $b_{i}=\tilde{\Phi}(e_{i1}\oplus 0)$ satisfies the required condition with $C_{M}=1$ .We note that $||(\varphi(a_{i}))_{i}||_{C}\leq||\varphi||_{\mathrm{c}\mathrm{b}}||(a_{i})_{i}||_{C}$ for any $\mathrm{c}\mathrm{b}$-map $\varphi$ and any finite sequence$(a_{\mathfrak{i}})_{\mathfrak{i}}$ . Hence the following is trivial.

Lemma 2.6. The property (P) inherits to a von Neumann subalgebra which is therange of a completely bounded projection.

As a corollary to Theorem 2.5, we see that a von Neumann subalgebra $M\subset \mathrm{B}(\mathcal{H})$

which is the range of a completely bounded projection is in fact injective. Weobserve that by the type decomposition and the Takesaki duality, it suffices toshow Theorem 2.5 for a von Neumann algebra of type $\mathrm{I}\mathrm{I}_{1}$ .

Let $M\subset \mathrm{B}(\mathcal{H})$ be a von Neumann algebra. An $M$ -central state is a state $\varphi$

on $\mathrm{B}(\mathcal{H})$ such that $\varphi(uxu^{*})=\varphi(x)$ for $u\in M$ and $x\in \mathrm{B}(\mathcal{H})$ (or equivalently$\overline{3\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}}$nomenclature is nonstandard.

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NARUTAKA OZAWA

$\varphi(ax)=\varphi(xa)$ for $a$ $\in M$ and $x\in \mathrm{B}(\mathcal{H}))$ . Recall that the celebrated theorem ofConnes states that a finite von Neumann algebra $M$ is injective iff there exists an$M$-central state $\varphi$ such that $\varphi_{|M}$ is a faithful normal tracial state.

Lemma 2.7. Let $M\subset \mathrm{B}(\mathcal{H})$ . Then, there exists an $M$ -central state if

$|| \sum_{i=1}^{n}u_{1}\otimes\overline{u_{i}}||_{\Re(\mathcal{H}\otimes\overline{\mathcal{H}})}=n$

for every $n$ and unitary elements $\mathrm{u}_{1},$ $\ldots$ , $u_{n}\in M$ .Proof. We first recall that $\overline{\mathcal{H}}$ is the complex conjugate Hilbert space of $\mathcal{H}$ and

$\overline{x}\in \mathrm{B}(\overline{\mathcal{H}})$ means the element associated with $x\in \mathrm{B}(\mathcal{H})$ . We have the canonicalidentification between the Hilbert space $\mathcal{H}\otimes\overline{\mathcal{H}}$ and the space $S_{2}(\mathcal{H})$ of the Hilbert-Schmidt class operators on $\mathcal{H}$ , given by $\xi\otimes\overline{\eta}rightarrow\langle\cdot,$

$\eta$) $\xi\in S_{2}(\mathcal{H})$ . Under thisidentification, $\sum a_{i}\otimes\overline{b_{i}}$ acts on $S_{2}(\mathcal{H})$ as $S_{2}( \mathcal{H})\ni hrightarrow\sum a_{i}hb_{i}^{*}\in S_{2}(\mathcal{H})$ .

Let $u_{1},$ $\ldots,$$u_{n}\in M$ be unitary elements such that $u_{1}=1$ . If 11 $\sum_{i=1}^{n}u_{i}\otimes\overline{u_{i}}||=n$ ,

then there exists a unit vector $h\in S_{2}(\mathcal{H})$ such that $|| \sum_{i=1}^{n}u_{i}hu^{*}\dot{.}||_{2}\approx n$ . Byuniform convexity, we must have $||u_{\mathfrak{i}}hu_{i}^{*}-h||_{2}\approx 0$ for every $i$ . This implies thatI $|u_{i}h^{*}hu_{i}-h^{*}h||_{1}\approx 0$ for every $i$ . It follows that $\varphi(x)=^{r}\mathrm{R}(h^{*}hx)$ defines a state on

$\mathrm{B}(\mathcal{H})$ such that $||\varphi\circ \mathrm{A}\mathrm{d}(u_{1})-\varphi||_{\hslash(?\{)}$ . $\approx 0$ for every $i$ . Therefore, taking appropriatelimit, we can obtain an $M$-central state. $\square$

Lemma 2.8 (Haagerup 1985). Let $M$ be a von Neumann algebra. Assume thatthere evists a constant $c>0$ with the following property; For every $n$ , unitaryelements $u_{1},$ $\ldots,$

$u_{n}\in M$ and every non-zero central projection $p\in M$ , we have

$|| \sum_{i=1}^{n}pu_{i}\otimes\overline{pw}||_{\mathrm{B}\mathrm{C}p\mathcal{H}\otimes\overline{p\mathcal{H}})}$ lii en.

Then, $M$ is injective.

Proof. Let $u_{1},$ $\ldots,$$u_{n}\in M$ be unitary elements and $p\in M$ be a non-zero central

projection. By assumption, we have

$||( \sum_{i=1}^{n}pu_{i}\otimes\overline{pu_{i}})^{k}||_{\mathbb{R}(p\otimes p\mathrm{i})}\mathcal{H}\urcorner\geq cn^{k}$

for every positive integer $k$ . Therefore, we actually have that

$|| \sum_{i=1}^{n}pu_{\mathfrak{i}}\otimes\overline{pu_{i}}||_{\dot{\mathrm{A}}}.\}(p\mathcal{H}\otimes p\neg \mathcal{H}\geq\lim_{karrow\infty}c^{1/k}n=n$ .

By Lemma 2.7, there exists a $pM$-central state $\varphi_{p}$ on $\mathrm{B}(\mathrm{p}M)$ for every non-zerocentral projection $p\in M$ . Fix a normal faithful tracial state $\tau$ on $M$ . For any finite

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SIMILARITY PROBLEMS AFTER PISIER

partition $\mathcal{P}=\{p_{i}\}_{\mathrm{i}}$ of unity by central projections in $M$ , we define the M-centralstate $\varphi_{P}$ on $\mathrm{B}(\mathcal{H})$ by

$\varphi_{P}(x)=\sum_{i}\tau(p_{i})\varphi_{p_{i}}(p_{i}xp_{i})$.

Taking appropriate limit of $\varphi_{P}$ , we obtain an $M$-central state $\varphi$ on $\mathrm{B}(\mathcal{H})$ such that$\varphi_{|M}=\tau$ . We conclude that $M$ is injective by Connes’s theorem. $\square$

For a finite sequence $(x_{i})_{i}$ in $\mathrm{B}(\mathcal{H})$ , we set

II $(x_{i})_{1}||_{oH}=|| \sum_{i}x_{i}\otimes\overline{x_{i}}||_{\mathrm{B}(\mathcal{H}\otimes\overline{\mathcal{H}})}^{1/2}$ .

We note that $||(x_{i})_{i}||_{oH}\leq||(x_{i})_{\mathfrak{i}}||_{R}^{1/2}||(x_{i})_{i}||_{C}^{1/2}\leq||(x_{i})_{i}||_{C\cap R}$ . Besides those appear-ing in Lemma 2.1, we have the following mysterious inequality (which manifeststhe self-dual property of the operator Hilbert spaces).

Lemma 2.9. For every finite sequences $(a_{i})_{i}$ in $\mathrm{B}(\mathcal{H})$ and $(b_{i})_{i}$ in $\mathrm{B}(\mathcal{K})$ , we have

$|| \sum_{i}a_{i}\otimes b_{i}||_{\mathcal{H}\otimes \mathcal{K}\rangle}\leq||(a_{i})_{i}||_{oH}||(b_{\mathfrak{i}}):||_{oH}$

Proof. We may assume that $\mathcal{K}=\overline{\mathcal{H}}$ and use $\overline{b_{i}}$ in the place of $b_{i}$ . Identifying $\mathcal{H}\otimes\overline{\mathcal{H}}$

with $S_{2}(\mathcal{H})$ as in the proof of Lemma 2.7, we see

$|| \sum_{:}a_{i}\otimes\overline{b_{i}}||_{\mathrm{B}(\mathcal{H}\otimes\overline{\mathcal{H}}\rangle}=\sup$ { $| \sum_{i}\mathrm{R}(ha_{i}kb^{*}.\cdot)|$: $h,$ $k\in S_{2}(\mathcal{H})$ with norm 1}.

Let $h,$ $k\in S_{2}(\mathcal{H})$ with norm 1 be given. Then, we can find decompositions $h=h_{1}h_{2}$

and $k=k_{1}k_{2}$ such that $h_{j},$ $k_{j}\in S_{4}(\mathcal{H})$ witf norm 1. It follows that

$| \sum_{:}\ulcorner \mathrm{R}(ha:kb_{i}^{*})|=|\sum_{i}\prime \mathrm{R}((h_{2}a_{i}k_{1})(k_{2}b_{i}^{*}h_{1}))|$

$\leq^{r}\mathrm{b}(\sum_{:}h_{2}a_{i}k_{1}k_{1}^{*}a_{\mathfrak{i}}^{*}h_{2}^{*})^{1/2r}\mathrm{R}(\sum_{i}h_{1}^{*}b_{i}k_{2}^{*}k_{2}b_{i}^{*}h_{1})^{1/2}$

$\leq||\sum_{i}a_{\mathfrak{i}}\otimes\overline{a_{i}}||_{i\mathrm{f}(H\emptyset\overline{\mathcal{H}})}^{1/2}||\sum_{i}b_{i}\otimes\overline{b_{i}}||_{\mathrm{B}(\mathcal{H}\otimes}^{1/2}\pi)$.

This proves the assertion. $\square$

Lemma 2.10. For every finite sequence ($x_{i}\rangle_{i}$ in $\mathrm{B}(\mathcal{H})$ , we have

11 $(x_{i})_{1}||_{C+R}\leq||(x_{i})_{i}||_{oH}$ .Proof. Let $\Phi:C\cap R\ni\theta_{i}->x_{i}\in \mathrm{B}(\mathcal{H})$ and take $z= \sum_{i}a_{i}\otimes\theta_{i}\in \mathrm{B}(\mathcal{H})\otimes(C\cap R)$ .We note that $||z||=||(a_{\mathfrak{i}})_{i}||_{C\cap R}\geq||(a_{i})_{i}||_{oH}$ . Hence, by Lemma 2.9, we have

$||( \mathrm{i}\mathrm{d}\otimes\Phi.)(z)||=||\sum a_{1}$. $\otimes x_{\mathfrak{i}}||\leq||(a_{i})_{\mathfrak{i}}||_{oH}||(x_{i}):||_{oH}\leq||(x_{\mathfrak{i}})_{i}||_{oH}||z||$ .This implies that $||(x_{i})_{i}||c+R=||\Phi||_{\mathrm{c}\mathrm{b}}\leq||(x_{i})_{\mathfrak{i}}||_{oH}$ . $\square$

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NARUTAKA OZAWA

We have prepared enough lemmas for the proof of Theorem 2.5.

Proof of Theorem 2.5. It is left to show that a finite von Neumann algebra $M$

with the property (P) is injective. To verify the assumption of Lemma 2.8, we giveourselves unitary elements $u_{1},$

$\ldots,$$u_{n}\in M$ , a non-zero central projection $p\in M$

and a constant $c>0$ such that$|1$ $(pu_{1})_{i}||_{oH}^{2}\leq cn$ .

Then, by Lemma 2.10 and the property (P), there exist $(a_{i})_{i}$ and $(b_{1})$ : in $M$ suchthat II $(a_{1})_{i}||c\leq C_{M}\sqrt{cn}$ , I $(b_{\iota’})_{i}||_{R}\leq C_{M}\sqrt{cn}$ and $pu_{i}=a_{i}+b_{i}$ for every $i$ . We fixa tracial state on $pM$ and denote by $||$ $||_{2}$ the corresponding 2-norm. It followsthat

$n= \sum_{i=1}^{n}||pu_{\mathfrak{i}}||_{2}^{2}\leq 2\sum_{i=1}^{n}(||a_{i},||_{2}^{2}+||b_{\mathfrak{i}}||_{2}^{2}\rangle\leq 2(||(a_{i},)_{1}||_{C}^{2}+||(b_{i})_{i}||_{R}^{2})\leq 2C_{M}^{2}rin,$ .

Therefore, we have $c\geq(2C_{M}^{2})^{-1}$ and we are done. $\square$

2.3. A characterization of nuclearity. Let $A$ be a (unital) C’-algebra. Wesay $A$ has the strong similarity property (abbreviated as (SSP)) if for every unitalcontinuous homomorphism $\pi:Aarrow \mathrm{B}(\mathcal{H})$ , there exists $S\in \mathrm{G}\mathrm{L}(\mathcal{H})\cap \mathrm{v}\mathrm{N}(\pi(A))$

such that $\mathrm{A}\mathrm{d}(S)\circ\pi$ is $\mathrm{a}*$-homomorphism.

Theorem 2.11 (Pisier 2005). A $O$ -algebra $A$ is nuclear iff it has the (SSP).

Proof. As we remarked, the ($‘ \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y}$ if” part follows from Diximier’s $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}+\mathrm{t}\mathrm{h}\mathrm{e}$

amenability of nuclear $\mathrm{C}^{*}$-algebra. To prove the “if” part, let $A$ be a C’-algebrawith the (SSP). By a standard direct sum argument, it is not hard to see thatthere exists a constant $C>0$ with the following property; Every unital continuoushomomorphism $\pi:Aarrow \mathrm{B}(\mathcal{H})$ with $||\pi||\leq 5^{4}$ , there exists $S\in \mathrm{G}\mathrm{L}(\mathcal{H})\cap \mathrm{v}\mathrm{N}(\pi(A))$

with $||S||||S^{-1}||$ $\leq C$ such that Ad(S) $\circ\pi$ is $\mathrm{a}*$-homomorphism. Let $A\subset \mathrm{B}(\mathcal{H})$ bea $\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{a}\mathrm{l}*$-representation. It suffices to show that $A’$ is injective. Let $(x_{\mathfrak{i}})_{i}$ be afinite sequence in $A’$ with $||(x_{i},)_{i}||_{C+R}\leq 1$ . Since $\mathrm{B}(\mathcal{H})$ is injective, there exist $(c_{i},)_{i}$

and $(d_{t}’)_{i}$ in $\mathrm{B}(\mathcal{H})$ such that $||(\mathrm{G})_{i}||_{C}\leq 1$ , II $(d_{i})_{i}||_{R}\leq 1$ and $x_{\mathfrak{i}}=c_{i}+d_{i}$ for every$i$ . We define a derivation $\delta:Aarrow \mathrm{B}(\mathcal{H})-\otimes \mathcal{L}\mathrm{F}_{\infty}$ by

$\delta(a)=\delta_{\Sigma c_{\ell\otimes\lambda(s)(a\otimes 1)=\sum_{i}\otimes \mathcal{L}\mathrm{F}_{\infty}}}‘\delta_{c_{i}}(a)\otimes\lambda(s_{i})\in \mathrm{B}(\mathcal{H})\otimes E_{\lambda}\subset \mathrm{B}(\mathcal{H})^{-}$ .

We recall from the proof of Theorem 2.3 that $\lambda(s_{i})=u_{i}+v_{i}$ with $||(u_{i})||_{C}\leq 1$ and$||(v_{i})||_{R}\leq 1$ . Since $\delta_{c_{i}}=\delta_{-d}$

‘ on $A$, we have $\delta=\delta_{B}$ , where $B= \sum(c_{i}\otimes v_{i}-d_{i}\otimes u_{i})$

4We can choose any other number that is strictly greater than 1 by scaling the 6 later.

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SIMILARITY PROBLEMS AFTER PISIER

with $||B||\leq||(c_{i})_{i}||c||(v_{i})||_{R}+||(d_{i})_{i}||_{R}||(u_{i})||_{C}\leq 2$. Hence, we have $||\delta||_{\mathrm{c}\mathrm{b}}\leq 4$. Wedefine a homomorphism $\pi:Aarrow \mathrm{M}_{2}(\mathrm{B}(\mathcal{H})^{-}\otimes \mathcal{L}\mathrm{F}_{\infty})$ by

$\pi(a)=(a\otimes 10$ $a\otimes 1\delta(a))$ .

By the assumption on the (SSP), there exists an invertible element $S\in \mathrm{v}\mathrm{N}(\pi(A))$

with $||S||||S^{-1}||\leq C$ such that Ad(S) $0\pi$ is a $*$-homomorphism. By the proofof Lemma 1.2, there exists $T\in \mathrm{B}(\mathcal{H})-\otimes L\mathrm{F}_{\infty}$ with $||T||\leq C^{2}$ such that $\delta(a)=$

$\delta_{\mathit{1}’},(a\otimes 1)$ . Let $Q:\mathcal{L}\mathrm{F}_{\infty}arrow E_{\lambda}$ be the projection appearing in Theorem 2.3. Since$\delta(A)\subset \mathrm{B}(\mathcal{H})\otimes E_{\lambda}$ and $\mathrm{i}\mathrm{d}\otimes Q$ is $A$-linear, we have

$\delta(a)=(\mathrm{i}\mathrm{d}\otimes Q)(\delta(a))=\delta_{(\mathrm{i}\mathrm{d}\otimes Q)(’\mathit{1}’)}(a\otimes 1)$

for every $a$ $\in A$ . We write $( \mathrm{i}\mathrm{d}\otimes Q)(T)=\sum z_{i}\otimes\lambda(s_{i})$ . Then, by Lemma 2.1 andTheorem 2.3, we have

$||(z_{i})_{i}||_{C\cap R}\leq||(\mathrm{i}\mathrm{d}\otimes Q)(T)||\leq||Q||_{\mathrm{c}\mathrm{b}}||T||\leq 2C^{2}$.Since $\lambda(s_{1})’ \mathrm{s}$ are linearly independent, we have $\delta_{c_{i}}=\delta_{z_{i}}$ , or equivalently $\mathrm{q}-z_{i}\in A’$ .Therefore, we have $a_{i}=\mathrm{q}-z_{i}\in A’$ with

$|\mathrm{I}$ $(a:)_{i}||_{C}\leq$ I $(c_{i})_{\mathfrak{i}}||_{C}+||(z_{i})_{i}||_{C}\leq 1+2C^{2}$ ,

and likewise $b_{i}=x:-a_{1}=d_{i}+z_{i}\in A’$ with $||(b_{\mathfrak{i}})_{i}||_{R}\leq 1+2C^{2}$ . We conclude theinjectivity of $A’$ by Theorem 2.5. $\square$

We say a group $\Gamma$ has the (SSP) if for every $\mathrm{u}.\mathrm{b}$ . representation $\pi:\Gammaarrow \mathrm{G}\mathrm{L}(\mathcal{H})$ ,there exists $S\in \mathrm{G}\mathrm{L}(\mathcal{H})\cap \mathrm{v}\mathrm{N}(\pi(\Gamma))$ such that Ad(S) $\circ\pi$ is a unitary representation.

Corollary 2.12. A discrete group $\Gamma$ is amenable iff it has the (SSP).

Proof. This follows from the fact that $\Gamma$ is amenable iff $C^{*}\Gamma$ is nuclear. $\square$

3. SIMILARITY LENGTH OF $\mathrm{C}^{*}$-ALGEBRAS

The following is the fundamental characterization of a homomorphism which issimilar to $\mathrm{a}*$-homomorphism. This has several applications to dilation theory.

Theorem 3.1 (Haagerup, Paulsen). Let $A$ be a unital C’-algebra (or just a unitaloperator algebra), $\pi:Aarrow \mathrm{B}(\mathcal{H})$ be a unital homomorphism and $C>0$ be aconstant. Then, $||\pi||_{\mathrm{c}\mathrm{b}}\leq C$ iff there exists $S\in \mathrm{G}\mathrm{L}(\mathcal{H})$ with $||S||||S^{-1}||\leq C$ suchthat $||\mathrm{A}\mathrm{d}(S)\circ\pi||_{\mathrm{c}\mathrm{b}}=1$ .Proof. The $‘(\mathrm{i}\mathrm{f}$

” part is obvious. To prove the “only if” part, let $A\subset \mathrm{B}(H)$ and$\pi:Aarrow \mathrm{B}(\mathcal{H})$ be a homomorphism with $||\pi||_{\mathrm{c}\mathrm{b}}\leq C$ . By a Stinespring typetheorem, there exist a Hilbert space $\hat{\mathcal{H}},$

$\mathrm{a}*$-homomorphism $\sigma:\mathrm{B}(H)arrow \mathrm{B}(\hat{\mathcal{H}})$ , andoperators $V\in \mathrm{B}(\mathcal{H},\hat{\mathcal{H}}),$ $W\in \mathrm{B}(\hat{\mathcal{H}}, \mathcal{H})$ with $||V||||W||\leq||\pi||_{\mathrm{c}\mathrm{b}}$ such that

$\forall a\in A$ $\pi(a)=V\sigma(a)W$.

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NARUTAKA OZAWA

Let $\mathcal{K}_{1}=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}(\sigma(A)W\mathcal{H})$ . The subspace $\mathcal{K}_{1}$ is $\sigma(A)$-invariant and we may assumethat $V=VP_{\mathcal{K}_{1}}$ . Since

$V\sigma(a)(\sigma(x)W\xi)=\pi(ax)\xi=\pi(a)V\sigma(x)W\xi$,

we have $V\sigma(a)P_{\mathcal{K}_{1}}=\pi(a)V$ for every $a$ $\in A$ . It follows that $\mathcal{K}_{2}=\mathrm{k}\mathrm{e}\mathrm{r}V\subset \mathcal{K}_{1}$ isalso $\sigma(A)$-invariant. Hence $\mathcal{L}=\mathcal{K}_{1}\ominus \mathcal{K}_{2}$ is $‘(\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}$-invariant” under $\sigma(A)$ , i.e.,

$\forall a\in A$ $P_{\mathcal{L}}\sigma(a)=P_{c\sigma}(a)P_{L}$ .Consequently, we have

$\forall a\in A$ $\pi(a)=VP_{c\sigma}(a)W=VP_{\mathcal{L}}\sigma(a)P_{\mathcal{L}}W$.Since $VP_{\mathcal{L}}$ is injective on $\mathcal{L}$ and $VP_{L}W=\pi(1)=1$ , the operator $S=VP_{L}$ is alinear isomorphism from $\mathcal{L}$ onto $\mathcal{H}$ with $S^{-1}=P_{\mathcal{L}}W$ . We have $\pi=\mathrm{A}\mathrm{d}(S)\circ\sigma$ with$||S||$ Il $S^{-1}$ Il $\leq C$ and, since $\mathcal{L}\cong \mathcal{H}$ , we are done. $\square$

Corollary 3.2. A derivation $\delta$ is inner iff it $\dot{u}$ completdy bounded.

By a standard direct sum argument, we obtain the following.

Corollary 3.3. Let $A$ be a unital $O$ -algebra with the $(\mathrm{S}\mathrm{P})$ . Then, there eansts afunction $f$ on $[1, \infty)$ such that

$||\pi||_{\mathrm{c}\mathrm{b}}\leq f(||\pi||)$

for every unital continuous homomorphism $\pi:Aarrow \mathrm{B}(\mathcal{H})$ .Deflnition 3.4. Let $A$ be a unital C’-algebra (or a unital operator algebra). Thesimilarity length of $A$ , denoted by $l(A)$ , is the smallest integer $l$ with the followingproperty; There exists a constant $C>0$ such that for any $x\in \mathrm{M}_{\infty}(A)$ , there exist$\alpha_{0},$ $\alpha_{1},$

$\ldots\alpha\iota\in \mathrm{M}_{\infty}(\mathbb{C})$ and $D_{1},$$\ldots,$

$D_{i}\in \mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}_{\infty}(A)$ satisfying$x=\alpha_{0}D_{1}\alpha_{1}\cdots D\iota\alpha_{i}$

and

$\prod_{m=0}^{i}||\alpha_{m}||\prod_{m=1}^{l}||D_{m}||\leq C||x||$ .

Here, $\mathrm{M}_{\infty}(A)=\bigcup_{n=1}^{\infty}\mathrm{M}_{n}(A)$ and $\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}_{\infty}(A)\subset \mathrm{M}_{\infty}(A)$ is the vet of diagonalmatrices with entries in $A$ . If there is no $l$ satisfying the above condition, then weset $l(A)=\infty$ by convention.

Theorem 3.5 (Pisier 1999). Let $A$ be a unital $\theta$ -algebra (or a unital operatoralgebra) with $\dim(A)>1$ . The following are equivalent.

(1) $A$ has the $(\mathrm{S}\mathrm{P})$ .(2) There erist $d>0$ and $C>0$ such that $||\pi||_{\mathrm{c}\mathrm{b}}\leq C||\pi||^{d}$ for every unital

continuous homomorphism $\pi:Aarrow \mathrm{B}(\mathcal{H})$ .(3) $l(A)\leq d$ .

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SIMILARITY PROBLEMS AFTER PISIER

The constant $d$ appearing in the conditions (2) and (3) are taken to be sameand are possibly non-integer. It follows that the “optimal” function $f$ appearingin Corollary 3.3 is a polynomial of degree $l(A)$ . The implication (2) $\Rightarrow(1)$ followsfrom Theorem 3.1. We do not prove the hard implication (1) $\Rightarrow(3)$ , but explain(3) $\Rightarrow(2)$ ;

$|| \pi(x)||=||\alpha_{0}\pi(D_{1})\alpha_{1}\cdots\pi(D_{i})\alpha_{l}||\leq||\pi||^{l}\prod_{m=0}^{l}||\alpha_{m}||\prod_{m=1}^{l}||D_{m}||\leq C||\pi||^{l}||x||$

for $x=\alpha_{0}D_{1}\alpha_{1}\cdots D\iota\alpha_{1}\in \mathrm{M}_{\infty}(A)$ .For a unital $\mathrm{C}^{*}$-algebra $A$ with $\dim(A)>1$ , it is known that(1) $l(A)=1\Leftrightarrow\dim(A)<\infty$ (Exercise),(2) $l(A)=2\Leftrightarrow A$ is nuclear with $\dim(A)=\infty$ (Pisier 2004),(3) $l(A)\leq 3$ if $A$ has no tracial state,(4) $l(M)=3$ if $M$ is a type $\mathrm{I}\mathrm{I}_{1}$ factor with the property $(\Gamma)$ (Christensen 2002),(5) $l(A)= \max\{l(I), l(A/I)\}$ for every closed 2-sided ideal $I\triangleleft A$ (Exercise).

It is not known whether there exists a unital C’-algebra with $l(A)>3$ . Wenote that an affirmative answer to Similarity Problem A would imply that thereexists $l_{0}$ such that $l(A)\leq l_{0}$ for every C’-algebra $A$ . We close this note by showing$l(A)\leq 3$ for any C’-algebra $A$ which contains a unital copy of the Cuntz algebra$O_{\infty}$ . (The case where $A$ has no tracial state is then dealt by passing to the seconddual.)

Let $x\in \mathrm{M}_{n}(A)$ be given. We choose unitary matrices $W_{1},$ $W_{2}\in \mathrm{M}_{n}(\mathbb{C})$ with$|W_{1}(i,j)|=|W_{2}(i,j)|=n^{-1/2}$ for all $i,j$ (e.g., $W_{k}(i,j)=n^{-1/2}\exp(2\pi\sqrt{-1}ij/n)$ ).Let $D_{1}(i)=S_{i}^{*}$ and $D_{3}(j)=S_{j}$ for every $i,j$ , where $s_{:}’ \mathrm{s}$ are isometries satisfying$S_{i}^{*}S_{j}=\delta_{i,j}I$ . For every $k$ , we set

$D_{2}(k)=n \sum_{i,j}\overline{W_{1}(i,k)}S_{i}x_{i,j}S_{j}^{*}\overline{W_{2}(k,j)}$

$=n$ $(\overline{W_{1}(1,k)}S_{1} \overline{W_{1}(n,k)}S_{n})(^{W_{2}}W_{2}(=_{:}^{(k,1)S_{1}}k, n)S_{n}^{*}’)$

Fhrom the latter expression, we see that $||D_{2}(k)||\leq||x||$ . We obtained $W_{1},$ $W_{2}\in$

$\mathrm{M}_{n}(\mathbb{C})$ and $D_{1},$ $D_{2},$ $D_{3}\in \mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}_{n}(A)\subset \mathrm{M}_{n}(A)$ such that

$||D_{1}||||W_{1}||||D_{2}||||W_{2}||||D_{3}||\leq||x||$

and$x=D_{1}W_{1}D_{2}W_{2}D_{\mathrm{S}}$ .

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NARUTAKA OZAWA

Indeed, we have

$(D_{1}W_{1}D_{2}W_{2}D_{3})_{i,j}= \sum_{k=1}^{n}S_{i}^{*}W_{1}(i, k)D_{2}(k)W_{2}(k,j)S_{j}$

$=n \sum_{k=1}^{n}|W_{1}(i, k)|^{2}|W_{2}(k,j)|^{2}x_{i,j}=x_{i,j}$ .

REFERENCES[1] G. Pisier, Introduction to operator space theory. London Mathematical Society Lecture Note

Series, 294. Cambridge University Press, Cambridge, 203.[2] –, Simdarity problems and completely bounded maps. Second, expanded edition. In-

cludes the solution to “The Halmos problem”. Lecture Notes in Mathematics, 1618. Springer-Verlag, Berlin, 2001.

[3] –, A similanty degree characterization of nudear $\sigma$ -algebras. $\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\dot{\mathrm{o}}\mathrm{t}$ .math. $\mathrm{O}\mathrm{A}/0409091$

[4] –, Simultaneous similarity, bounded generation and amenability. Preprint.math. $\mathrm{O}\mathrm{A}/0508223$

DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY op TOKYO, KOMABA, 153-8914$E$-mail address: narutakaQms. $\mathrm{u}$-tokyo. $\mathrm{a}\mathrm{c}$ . jp

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